arX

iv:0

801.

1017

v1 [

astr

o-ph

] 7

Jan

2008

An implicit numerical algorithm for solving thegeneral relativistic hydrodynamical equations around

accreting compact objects

A. Hujeirat∗, M. Camenzind and B.W. Keil

ZAH, Landessternwarte Heidelberg-Konigstuhl, Universitat Heidelberg, 69117Heidelberg, Germany

Abstract

An implicit algorithm for solving the equations of general relativistic hydrodynamicsin conservative form in three-dimensional axi-symmetry ispresented. This algorithm is adirect extension of the pseudo-Newtonian implicit radiative magnetohydrodynamical solver-IRMHD- into the general relativistic regime.

We adopt the Boyer-Lindquist coordinates and formulate thehydrodynamical equationsin the fixed background of a Kerr black hole. The set of equations are solved implicitlyusing the hierarchical solution scenario (HSS). The HSS is efficient, robust and enables theuse of a variety of solution procedures that range from a purely explicit up to fully implicitschemes. The discretization of the HD-equations is based onthe finite volume formulationand the defect-correction iteration strategy for recovering higher order spatial and temporalaccuracies. Depending on the astrophysical problem, a variety of relaxation methods canbe applied. In particular the vectorized black-white Line-Gauss-Seidel relaxation methodis most suitable for modeling accretion flows with shocks, whereas the Approximate Fac-torization Method is for weakly compressible flows.

The results of several test calculations that verify the accuracy and robustness of the algo-rithm are shown. Extending the algorithm to enable solving the non-ideal MHD equationsin the general relativistic regime is the subject of our ongoing research.

Key words: plasmas, (magnetohydrodynamics:) MHD, gravitation, relativity, shockwaves, methods: numericalPACS:95.30.Qd, 95.30.Sf

∗ Corresponding author. Tel.:++49 6221 5417 63, fax:+49 6221 5417 02, Address: ZAH,Landessternwarte, Konigstuhl 12, D-69117 Heidelberg, Deutschland

Email addresses:A.Hujeirat@lsw.uni-heidelberg.de (A. Hujeirat),M.Camenzind@lsw.uni-heidelberg.de (M. Camenzind),B.Keil@lsw.uni-heidelberg.de (B.W. Keil).

Preprint submitted to New Astronomy 3 November 2018

1 Introduction

The field of astrophysical fluid dynamics (henceforth AFD) deals with the macro-scopic evolution of gaseous-matter and plasmas in a wide variety of circumstancesin astrophysics. The scope of AFD is broad, encompassing topics such as star for-mation, accretion phenomena onto compact and young stellarobjects, dynamicsof the interstellar medium, jets, winds and outflows emerging from around youngstellar objects, from quasars and microquasars, supernovae explosion,γ−ray burstsand structure formation in the universe.

One of the ultimate aims of numerical astrophysics is to develop a black box al-gorithm which contains numerical solvers that are unconditionally stable, robust,efficient, Newtonian, relativistic and capable of treating flows that are strongly com-pressible, weakly incompressible, self-gravitating, radiating, magnetized multi-com-ponent-plasmas with high spatial and temporal accuracies on unstructured meshesand to provide the required solutions within the scale of hours. While this goal isunlikely to be achieved within the next few years, the increasing number of sophis-ticated numerical algorithms developed during the last twodecades is remarkablyencouraging. In particular, significant improvements havebeen achieved in increas-ing the spatial dimensions and enhancing the efficiency and accuracy of numericalalgorithms (see e.g. Nagel et al., 2006).On the other hand, the problem of robustness of the solvers inAFD has been barelyconsidered nor even seriously discussed. In this paper we discuss the robustnessproblem in AFD, present enhancement strategies and addressthe necessity of con-structing robust general relativistic implicit and radiative MHD solvers. For com-pleteness we review the concepts of efficiency and robustness of numerical solversin computational fluid dynamics.

A numerical solver is said to be relatively efficient if the corresponding number ofalgebraic manipulations per time step per computer-processor can be made respec-tively small. As a consequence, using high performance computers with large num-ber of computing processors, a significant progress has beenachieved in improv-ing runtime efficiencies, provided the computing load is distributed appropriately.Thus, by means of modern hardware technology, the efficiency of computer-codescould be enhanced even without modifying the employed mathematical approach.This has led to uncoordinated developments of relatively large number of computer-codes, that may differ in efficiency and integrated physical processes but are almostessentially identical with respect to the mathematical solution procedure. This phe-nomenon can be attributed to the lack of robustness. By robustness, we mean thecapability of a solver to be applied to a large class of problems without modifyingthe algorithmic core significantly.

In an attempt to enhance both efficiency and robustness, the hierarchical solutionstrategy (HSS) has been suggested (Fig. 1 and 2, also see Hujeirat (2005) and the

2

references therein). The HSS relies on the fact that any set of hydrodynamical, mag-netohydrodynamical or radiative equations are linearize-able and therefore can bere-written in a simple matrix equationAq= b, whereA, q, b are the coefficient ma-trix, vector of unknown variables and the vector of known quantities, respectively.Applying the defect-correction strategy (Stetter, 1978; Bohmer, Stetter, 1984), wemay then re-write this equation as:Aδq = d,whereδq andd denote respectively thevectors of small-time corrections and the defect, providedd is consistent with thereal mathematical equations by construction. The matrix A in the latter equationcan be replaced by a variety of matrices that correspond to a sequence of numericalapproaches that ranges from purely explicit to strongly implicit (Hujeirat, 2005).In this formulation, explicit methods arise as a special case, in which A is beingreplaced by the most easiest invertible-matrix: the identity matrix I. Based on thisformulation, the Courant condition follows from the requirement that the matrix Ashould be stable-invertible.

Therefore, strongly implicit and explicit methods are different variants of the samealgebraic problem. While the former retain almost all the entries of the matrix Ain the inversion procedure, the latter rely on neglecting all off-diagonal entries aswell as crudely simplifying the diagonal elements. These methods are well-unifiedwithin the HSS, and that depending on the physical properties of the flow, a direc-tive will carefully select the entries of A that are relevantfor the problem.

In Table 1 we have summarized properties of several numerical methods. Thus,as long as efficiency is concerned, explicit methods are unrivalled candidates, pro-vided the flows are strongly time-dependent and compressible. However, due to therelatively large sound speed, these methods may stagnate both in modeling incom-pressible or even weakly compressible flows. To clarify thispoint, we mention thatthe time step-size in explicit methods must satisfy the Courant-Friedrichs-Lewy(CFL) condition:

δtexp< min{ ∆xU + Vs

} = min{ ∆xU(1+ 1/M)

},

where the minimum function runs over all points of the domainof calculation. Hereδtexp,∆x,U,Vs,M correspond to the explicit time step-size, space increment, veloc-ity, sound speed and the Mach number, respectively. Therefore, asM→ 0 the flowbecomes incompressible and the time step size approaches zero; hence a stagnationof the time-advancement procedure. We note that although inthis case using con-sistent implicit solution procedures is necessary, by no means it is sufficient. Hereit has been verified that standard implicit solvers experience serious difficulties insimulating low Mach number flows, typically found in the interior layers of stars,planets as well as around moving vehicles in the Earth atmosphere.

The above discussion addresses the following questions: 1)Relativistic fluid mo-tions typically occur on the dynamical time scale. The advantages of still using im-

3

plicit solvers should be clarified. 2) Multigrid methods have been shown to displayconvergence which depends weakly on the number of unknowns in the finite space.In combination with nested iteration, the multigrid methodcan solve these prob-lems to truncation error accuracy in a number of operations that is proportional tothe number of unknowns. Therefore the reason for still favouring the prolongationstrategy over multigrid or adaptive mesh refinement needs tobe explained. 3) Thestorage capacities of modern computers to date are capable of handling the entriesof large matrices that correspond to the 3D MHD equations. Thus, the reliabilityand credibility of 3D axi-symmetric algorithms should be justified.

In fact, there are several reasons that justify using implicit numerical proceduresfor modeling relativistic flows. In particular:

• The set of relativistic MHD equations is generically a highly coupled non-linearsystem, which gives rise to fast growing perturbations due to non-linearities,thereby imposing a further restriction on the size of the time step.

• In most of the cases the horizon of a black hole represents a geometrical singu-larity. The deformation of the geometry grows non-linearlywhen approachingthe black hole. Thus, in order to capture flow-configurationsin the vicinity ofa black hole accurately, a non-linear distribution of the grid points is necessary,which, again, may destabilize explicit schemes.

• Depending on the evolutionary conditions, non-relativistic flows may becomeultra-relativistic or vice versa. However, almost all non-relativistic astrophysicalflows known to date are considered to be dissipative and diffusive. Therefore, inorder to track their time-evolution reliably, the employednumerical solver shouldbe capable of treating the corresponding second order viscous terms properly.

• The timescales of most astrophysical flows are considered tohave a great dispar-ity. Stability requirement of conditionally stable methods however requires thatthe time step size should be a small fraction of the shortest possible timescale.This implies that, in order to cover a timescale of astrophysical relevance, anextremely large number of time steps would be required, which would give riseto prohibitive computational costs. Furthermore, the accumulated round off er-rors resulting from performing a large number of time-extrapolations for time-advancing a numerical hydrodynamical solution may easily cause divergence.

• The initial conditions of most astrophysical phenomena arenot known. There-fore, in carrying out global hydrodynamical simulations, the end solution shouldweakly depend on the initial flow configuration. Conditionally stable numericalmethods, however, rely on time-advancing of the initial conditions.

The latter reason may explain also why using the prolongation strategy is advan-tageous over classical multigrid. Worth noting is that the main building blocks ofmultigrid methods are:

(1) Restriction, i.e., down sampling of the residual errorsinto coarser meshes.(2) Residual smoothing: reducing the high frequency errorsby performing several

4

iterations, using a computationally efficient iterative procedure such as Jacobior Gauss-Seidel.

(3) Prolongation, which relies on the interpolation from the coarse onto finermeshes.

The high-frequency errors here are reduced by cheap smoothing on the fine meshes,whereas the low-frequency errors are reduced by defect correction on the coarsermeshes. As the bulk of the algebraic operations are made on the coarser meshes, thecombined solution procedure is considerably efficient. However, multigrid meth-ods display satisfactory convergence, if the underlying flows are predominantlydiffusion-dominated. In the case of advection-dominated flows,errors, that are re-sponsible for the slow convergence on the fine meshes, can be easily advected bythe flow on the coarser meshes, thereby reducing the coarse grid correction. In thecase of astrophysical flows, the corresponding equations may change their char-acter from Newtonian into ultra-relativistic or vice versa. Unlike Newtonian flowsthat are generically diffusion-dominated, relativistic flows may become predomi-nantly advection-dominated, depending on how large the corresponding Lorentzfactors are. Hence, multigrid methods may fail to provide the expected rate of con-vergence.

Finally we note that in order to model the formation and acceleration of relativisticflows in the vicinity of ultra-compact objects accurately, it is necessary to cover thedomain of calculation by a strongly stretched mesh. Furthermore, Lorentz factorsenhance the inner-coupling of the relativistic equations and give rise to a largerspectrum of non-linearities. These numerical difficulties combined with the need toinclude sophisticated magnetic and radiative processes make the construction of afully 3D algorithm, at the moment, a computationally unrealizable numerical task.

Therefore, in this paper, we do not intend to perform 3D global simulations, butrather focus on the algorithmic structure of unconditionally stable and robust 3Daxi-symmetric solvers. These algorithms should enable us to search for station-ary or quasi-stationary solutions for the fully-coupled radiative MHD equations inwhich detailed physical processes are taken into account.The paper runs as follows: in Sec. 2 we describe the relativistic hydrodynamicalequations, in Sec. 3 the transformation between the primitive and conservativevariables is described. The numerical solution and the discretization methods arepresented in Sections 4 and 5. In Sec. 6 we present the resultsof several test calcu-lations and end up with a summary in Sec. 7.

2 The hydrodynamical equations in Kerr spacetime

In the present study we intend to numerically solve the equations of hydrodynamicsin both ultra-relativistic and purely Newtonian regimes. Unlike the usual conven-

5

Explicit Implicit HSS

solutionmethod

qn+1 = qn + δt dn qn+1 = qn + δtA−1d∗ qn+1 = αqn + (1− α)δtA−1d d∗

Type of flows

Strongly time-dependent,compressible,weakly dissipativeHD and MHDin 1, 2 and 3 dimen-sions

Stationary,quasi-stationary,highly dissipative,radiative andaxi-symmetricMHD-flows in 1, 2and 3 dimensions

Stationary,quasi-stationary,weakly compressible,highly dissipative,radiative andaxi-symmetric MHD-flows in 1, 2 and 3 di-mensions

Stability conditioned unconditioned unconditioned

Efficiency 1 (normalized/2D) ∼ m2 ∼ m2d

Efficiency:Enhancementstrategies

ParallelizationParallelization,preconditioning,multigrid

HSS, parallelization,preconditioning,prolongation

Robustness:Enhancementstrategies

i. subtime-steppingii. stiff termsare solvedsemi-implicitly

i. multiple iterationii. reducing the timestep size

i. multiple iterationii. reducing the timestep size, HSS

NumericalCodesNewtonian

Solvers1a

ZEUS&ATHENAb,

FLASHc, NIRVANA d,

PLUTOe, VAC f

Solver2g IRMHDh

NumericalCodesRelativistic

Solvers3i

GRMHD j , ENZOk,

PLUTOl , HARMm,

RAISHINn, RAMo,

GENESISp, WHISKYq

Solver4r GR-I-RMHDs

Table 1A list of only a part of the grid-oriented codes in AFD and their range of applications.

The matrix-equations in the first row are illustrated in Sec.4. In these equations,qn,n+1,δt, A, α andd∗ denote the vector of variables from the old and new time levels, time stepsize, a preconditioning matrix, a switch on/off parameter and a time-modified defect vector,respectively. “m” in row 4 denotes the bandwidth of the corresponding matrix.aBodenheimer et al. (1978); Clarke (1996),bStone, Norman (1992); Gardiner, Stone(2006), cFryxell et al. (2000),dZiegler (1998),eMignone, Bodo (2003); Mignone et al.(2007), f Toth et al. (1998),gWuchterl (1990); Swesty (1995),hHujeirat (1995, 2005);Falle (2003),iKoide et al. (1999); Komissarov (2004); Anninos et al. (2005); Meliani et al.(2007); Del Zanna et al. (2007),jDe Villiers, Hawley (2003), kWang et al. (2007),lMignone et al. (2007),mGammie et al. (2003),nMizuno et al. (2006),oZhang, MacFadyen(2006),pAlay et al. (1999),qBaiotti et al. (2003),rLiebendorfer et al. (2002),sthe presentalgorithm.

6

tion, in which the speed of light and the gravitational constant are set to unity, weuse the sound speed as the basic measure for velocities. Thisis reasonable as theradial motion of rotating flows around a compact object can beas low as 10−5±1 thespeed of light, whereas it is 10−2±1 of the sound speed. Close to the event horizon,all velocities become quantitatively comparable. This scaling enables the presentalgorithm to capture the structure of slow flows accurately and renders the appear-ance of terms that are extremely large or small due to scalingeffects. Additionally,the present solution procedure is actually an extension of the purely Newtoniansolver, IRMHD, into the general relativistic regime.

2.1 The metric

For completeness, we develop here the equations of hydrodynamics in the back-ground of spacetime metric of a Kerr black hole, using the Boyer-Lindquist coordi-nates (t, r, θ, φ). Adopting the 3+1 split of spacetime, a line element with the metricsignature (−,+,+,+) can be written as follows:

ds2 = −α2dt2 + hik(dxi + βidt)(dxk + βkdt), for j,k=1, 3. (1)

For the Kerr metric, the line element reads:

ds2 = −(α2 − βφβφ)dt2 + 2βφdφdt+ hikdxidxk, (2)

which corresponds to a matrix of the following entries:

gµν =

gtt 0 0 gtφ

0 grr 0 0

0 0 gθθ 0

gφt 0 0 gφφ

=

βφβφ − α2 0 0 βφ

0 hrr 0 0

0 0 hθθ 0

βφ 0 0 hφφ

. (3)

7

The coefficientsgµν in the Boyer-Lindquist coordinates and their related functionsare defined as follows:

∆ = r2 − 2rgr + a2

ρ2 = r2 + a2 sin2 θ

Σ2 = (r2 + a2)2 − a2∆ cos2 θ

ω = Σρ

cosθ

α =ρΣ

√∆

βr = βθ = 0, βφ = −ωc = −2aMBHr

cΣ2

Υ =ρ2Σ2

∆cos2 θ

hrr =ρ2

∆, hθθ = ρ2, hφφ = ω2

√−g = ρ2 cosθ = α√Υ.

(4)

c, MBH, G, rg(= GMBHc2 ), α, β and “a” denote the speed of light, mass of the black

hole, the gravitational constant, the gravitational radius, the lapse and shift func-tions and the Kerr-spin parameter, respectively. In writing these expressions, wemade use of the coordinate transformationθ = π/2− θ, where we use the latitudeθinstead of the polar distance angleθ; hence the appearance of ”cos” instead of ”sin”in the metric terms.

2.2 The governing equations

Following the internal energy formulation of Wilson (1972)and Hawley et al. (1984a,b),we derive the hydrodynamical equations from the four-velocity uµuµ = −c2, theconservation of baryonic number∇µ(ρuµ) = 0, the parallel component of the stress-energy conservation equationuν∇µTµν = 0 (to derive the internal energy equation)and from the transverse components (gξν + uξuν)∇µTµν = 0 (to derive the momen-tum equations).For viscous flows, the stress energy tensor reads:

Tµν = TµνPF+{

TµνVis

}

= ρ h uµuν + P gµν +

{

−η[σµν + Θ3

hµν]

}

, (5)

where TµνPF, TµνVis denote the stress energy tensor due to perfect and viscous flows,respectively. P,η, Θ, are the pressure, which is calculated from the equation ofstate corresponding to polytropic or to an ideal gas, the dynamical viscosity whichis assumed to be identical to the shear viscosity, andΘ (� ∇µuµ) which measures thedivergence or convergence of the fluid world lines, respectively. hµν = uµuν + gµν

8

is the spatial projection tensor, whereas ¯σ corresponds to the symmetric spatialshear tensor: ¯σµν = ∇ςuµhςν + ∇ςuνhςµ. In the case of an ideal gas, the pressure andenthalpy read:

P = (γ − 1)CVρT = (γ − 1)ρE

h = c2 + E + P/ρ = c2 + γE,(6)

whereγ, CV andE denote the adiabatic index, specific heat and internal energymeasured in the local rest frame of the fluid. For clarity, we re-write the hydrody-namical equations in flux conservative form:

∂D∂t+

1√−g

∂

∂r(√−gDVr) +

1√−g

∂

∂θ(√−gDVθ) = 0 (7)

∂Mr

∂t+

1√−g

∂

∂r(√−gMrV

r) +1√−g

∂

∂θ(√−gMrV

θ) =

− ∂P∂r− 1

2(MµMν

Mt)∂gµν∂r+ L2r

vis (8)

∂Mθ∂t+

1√−g

∂

∂r(√−gMθV

r) +1√−g

∂

∂θ(√−gMθV

θ) =

− ∂P∂θ− 1

2(MµMν

Mt)∂gµν∂θ+ L2θvis (9)

∂Mφ∂t+

1√−g

∂

∂r(√−gMφV

r) +1√−g

∂

∂θ(√−gMφV

θ) = L2φvis (10)

∂Ed

∂t+

1√−g

∂

∂r(√−gEdVr) +

1√−g

∂

∂θ(√−gEdVθ) =

− (γ − 1)Ed [∂ut

∂t+

1√−g

∂

∂r(√−gutVr) +

1√−g

∂

∂θ(√−gutVθ)

+ Φ + Γ − Λ, (11)

where D= ρut is the modified relativistic mass density. Mµ are the four-momenta:(M t,Mr,Mθ,Mφ) � D(ut, ur, uθ, uφ), whereD � Dh, andut is the time-like velocity,Vµ = uµ/ut is the transport velocity.L2ξvis are the spatial projections of the viscousstress energy tensor TµνVis (see Eq. 5) in the respective direction. These are obtainedfrom the projection of the viscous tensor along the vector normal to the hyperspace,i.e., constant in time:

L2ξvis = ∇µTµξVis = ∂µTµξVis + Γ

ξµλT

µλVis,

9

whereξ = {r, θ, ϕ}.∇µ corresponds to the spatial divergence of a tensor taken in theBoyer-Lindquist coordinates andΓξµλ are the Christoffel’s symbols of the secondkind.From the collection of the numerous viscous terms, we only consider the dominantsecond order operators, that are set to degenerate into the classical non-relativisticformulation of the Navier-Stokes equations if the sound speed becomes smallerthan a critical value (Tassoul, 1978).The viscosity coefficient here is based on the alpha-turbulent description,αt, mod-ified to respect causality. Hence the dynamical viscosity reads:

η = ρ < l >< vturb >= ρ < l > [αVs/ut] =

D(ut)2

[α < l > Vs] =D

(ut)2νt, (12)

where<> denotes a turbulent mean, vturb is the relativistic turbulent velocity andνtthe relativistic turbulent velocity coefficient andαt is a constant of order unity.

Equation (11) describes the time-evolution of the relativistically modified internalenergyEd = D CV T, where T is the plasma temperature.Φ, Γ, Λ correspondto heat function due to turbulent dissipation, other heating and cooling functions,respectively. Using the transformation qµ = gµνqν, we may define the transportvelocities Vµ as follows (see Hawley et al., 1984a,b):

ur = grr ur = grr ut (Vr/c)

uθ = gθθ uθ = gθθ ut (Vθ/c)

ut = gtt ut + gtφ uφ = ut [gtt + gtφ (Vφ/c)]

uφ = gtφ ut + gφφ uφ = ut [gtφ + gφφ (Vφ/c)].

(13)

The corresponding relativistic 4-momenta then read:

Mt = Dut� D

Mr = D[Vr/c]

Mθ = D[Vθ/c]

Mφ = D[Vφ/c],

(14)

10

from which the covariant 4-momenta may be obtained:

Mt = gtt Mt + gtφ Mφ

Mr = grr Mr

Mθ = gθθ Mθ

Mφ = gφφ Mφ + gφt Mt.

(15)

We note that by using this formulation of the HD-equations incombination with fi-nite volume discretization, mass and momenta are conservedup to small discretiza-tion errors. This is necessary in order to assure that inflowing non-rotating matterwould not gain angular momenta though it must rotate in the ergosphere.

2.3 Non-dimensional formulation

The algorithm presented here should be capable of modeling the time-evolutionof hydrodynamical flows both in the non-relativistic as wellas in the extreme-relativistic regimes. In order to avoid the appearing of extremely small coefficientsin the equations, the scaling variables listed in Table (2) are adopted.

Scaling variables Example (supermassive BH)

Mass: M 3× 108M⊙

Accretion rate: ˜M 10−2 MEdd

Distance: R 3RS

Temperature: T 108 K

Velocities: V VS

Density: ρ 10−12 g cm−3

Table 2Scaling variables that might be used to re-write the equations in non-dimensional form. Inthis tableRS (= 2rg) stands for the Schwarzschild radius (rg is the gravitational radius) andVS is the sound speed.

We now introduce the following additional non-dimensionalparameters:

{

ǫBH = rg/R, ǫ1 =VSVφ, ǫ6 =

VSc , a = a(rg/R)

}

. (16)

where a is the black hole spin.

11

The normalization of the 4-velocity and momentum yields:

(U t)2

[

gtt + 2

(

ǫ6

ǫ1

)2

gtφVφ + ǫ6

2grr (Vr )2 + ǫ6

2gθθ(Vθ)2 +

(

ǫ6

ǫ1

)2

gφφ(Vφ)2

]

= −1, (17)

and

gtt M2t + 2

(

ǫ6

ǫ1

)3

gtφMtMφ + ǫ62grr M2

r + ǫ62gθθM2

θ +

(

ǫ6

ǫ1

)2

gφφM2φ = −D

2, (18)

where

gtt = −1/α2

gtφ = a r/(αΣ)2

gφφ = [1 − ( 4α2 )(a r

Σ2 )]

grr = 1/grr = ∆/ρ2

gθθ = 1/gθθ = 1/ρ2.

(19)

We may write the equations of hydrodynamics in non-dimensional formulation in amanner that they smoothly adopt the Newtonian form in the non-relativistic regime:

∂D∂t+

1√−g

∂

∂r

(√

−ggrr

DU

)

+1√−g

∂

∂θ

(√

−ggθθ

DV

)

= 0 (20)

∂Mr

∂t+

1√−g

∂

∂r

(√

−ggrr

MrU

)

+1√−g

∂

∂θ

(√

−ggθθ

MrV

)

=

− ∂P∂r− 1

2

(MµMν

M t

)

∂gµν∂r+Qr

vis (21)

∂Mθ∂t+

1√−g

∂

∂r

(√

−ggrr

MθU

)

+1√−g

∂

∂θ

(√

−ggθθ

MθV

)

=

− ∂P∂θ− 1

2

(MµMν

M t

)

∂gµν∂θ+Qθvis (22)

∂Mφ∂t+

1√−g

∂

∂r

(√

−ggrr

MφU

)

+1√−g

∂

∂θ

(√

ggθθ

MφV

)

= Qφvis (23)

∂Ed

∂t+

1√−g

∂

∂r

(√

−ggrrEdU

)

+1√−g

∂

∂θ(

√

−ggθθEdV) =

− (γ − 1)Ed

[

∂ut

∂t+

1√−g

∂

∂r

(√

−ggrr

utU

)

+1√−g

∂

∂θ

(√

−ggθθ

utV

)]

12

+ Φ + Γ − Λ, (24)

where U= Vr √grr ,V = Vθ√

gθθ,Vφ = Vφ√

gφφ, h = 1+ ( ǫ26γ−1) T.

For flows approaching rotating black holes, the angular momentum is defined as:Mφ = D

√gφφ (Vφ + VφFDE), whereVFDE denotes the rotation of the flow that is

induced due to the frame-dragging effect:VφFDE = βφ/√

gφφ.Note that the radial velocity in this formulation approaches the speed of light as thematter crosses the event horizon.

3 The primitive variables

The above set of equations describes the time-evolution of the conserved quantitiesD,Mi andEd.However, the equation of state, the rate of transport, the applied work,cooling and heating rates are functions of essentially the primitive variablesρ,Vi

andT.Since the relation between the primitive and conservative variables is rather non-linear, an iterative solution procedure should be employed.

We note that the 4-momenta must satisfy the normalization condition: MµMµ =

−M2� −D

2. This is equivalent to solve the following equation forMt:

MµMµ = gµνMµMν = gtt M2

t + 2gtφMtMφ + grr M2r + gθθM2

θ + gφφM2φ = −D

2. (25)

Taking into account that the quantitiesMr , Mθ, Mφ are known at the end of eachtime step, we may substitute them in Eq. (24) to obtain a quadratic equation forMt,i.e.,

ℵM2t + iMt + ג = 0, (26)

whereℵ = gtt, i = 2gtφMφ and ג = grr M2r + gθθM2

θ + gφφM2φ.

Having obtainedMt, the contravariant quantityMt can be computed using the trans-formation:Mt = gttMt +gtφMφ, whereas the global Lorentz factor is obtained from:ut = Mt/Dh. Using Equation (24), the pressure can be calculated then from therelation:P = (γ − 1)Ed/ut.

4 The hierarchical solution strategy - HSS

The set of hydrodynamical equations are solved within the context of the hierar-chical solution strategy (HSS, see Hujeirat (2005)). HSS isbased on constructing

13

Fig. 1. A schematic description of the hierarchical clustering of the coefficient matrix em-ployed by the algorithm. Here, a cluster of coefficients is computed in the very early stage.The matrix-generator then selects the entries to be used forconstructing the matrix co-efficient appropriate for the solution procedure. Depending onthe matrix used, the solu-tion method may range from purely explicit to fully implicit. Interchange between solutionmethods is possible, as modifying, adding or removing entries is directly maintainable.

a coefficient matrixA, which results from linearizing the complete set of equationsin a fully implicit manner. Noting that the conservative formulation of the HD-equations yields a matrix coefficient that is highly sparse, it is reasonable to designa procedure which selects the entries for constructing the approximate matrixAmost appropriate for the flow problem. Depending on the structure of A, a suitable

14

e

A=

d

T ,

Tp

µR

MH

D

Eν

RT

E:

Nex

t tim

e st

ep

?A

noth

er

itera

tion N

ext

time

step

?

?

Nex

t tim

e st

ep

Ano

ther

5

itera

tion

Nex

t tim

e st

ep

MA

TG

EN

2

MA

TG

EN

3

-

KS

IM-

A q

= b

A q

= b

A q

= b

A q

= b

A q

= b

A q

= b

A q

= b

III

III SP

SP

tot

MH

D MH

D

HD

HD

A q

= b

SP

S

P

SP

A

q =

b1 2 3 4 5

IOSH

D

MH

D

SP

tot

tot

1

A q

= b

2 3 4 5

1 2 3 4

Fig. 2. A schematic description of the hierarchical structure of the algorithm. Stage I cor-responds to the implicit operator splitting approach (IOS), which is most appropriate forfollowing the early time-dependent phases of the flow. The solutions obtained can then beused as initial conditions for Stage II, where the hydro-equations are solved as a singlecoupled system, followed by the magneto component, which isagain solved as a singlecoupled system. Here, high spatial and temporal accuraciesin combination with the pro-longation/restriction strategy may be used. Similarly, the solutionsobtained in this stagemay be used as starting solutions for Stage III, where steadysolutions for the fully coupledset of equations consisting of the zero moment of the radiation field and the MHD equationsare sought. In this stage, pre–conditioned Krylov sub–space iterative methods are consid-ered to be robust and efficient. The very last stage, Stage IV, corresponds to the casewheresolutions for the internal energy equations weakly coupledwith the 5D radiative transferequation are sought.

15

iterative method within the context of defect-correction method may be employedto assure consistency and convergence.

For example, if we want to simulate a two-dimensional weaklycompressible, non-magnetized and non-radiating flow between two concentric spheres, then the above-mentioned procedure is set to select the entries from the cluster of coefficients thatcorresponding just to the equations to be solved (see Fig. 1), which are used thento construct the preconditioningA.

To clarify the procedure, we re-write the set of equations ina conservative vectorform:

∂~q∂t+ Lr,rr ~F + Lθ,θθ ~G = ~f , (27)

where ~F and ~G are fluxes ofq, andLr,rr, Lθ,θθ are first and second order operatorsthat describe the advection and diffusion of the vector variables~q in r andθ direc-tions. ~f corresponds to the vector of source functions.

In order to enhance the mathematical consistency and increase the spatial and tem-poral accuracies of the numerical scheme without a substantial increase of the com-putational costs, we adopt the so-called prediction-correction iteration procedure.Therefore, we re-write Eq. (27) in residual form:R = ∂~q

∂t + Lr,rr ~F + Lθ,θθ ~G − ~f = 0,and adopting a five star staggered grid discretization, we may apply the Newton-linearization to calculate the Jacobian,Jm1,n1 �

∂Rm1∂qn1

, where m1, n1 are integers thatrun over the number of equations and variables. The solutioncan be obtained thenas follows:

~qi+1 = ~qi − J−1m1,n1R

i ,

wherei is the iteration level. By inspection of the Jacobian J, it can be easily verifiedthat it has the following block matrix structure:

δqj,k

δt+ Srδqj−1,k + Drδqj,k + S

rδqj+1,k

+ Sθδqj,k−1 + Dθδqj,k + Sθδqj,k+1 = RHSn

j,k, (28)

whereδq = qi+1 − qi , and which, in the linear case, reduces to time-difference ofq. The subscripts “j” and “k” denote the grid-numbering in ther andθ directions,respectively, andRHSn = [ ~f − Lr,rr ~F − Lθ,θθ ~G]n. Sr,θ andS

r,θmark the sub-diagonal

and super-diagonal block matrices, respectively.Dr,θ corresponds to diagonal block

16

matrices.To outline the directional dependence of the block matrices, we re-write Eq. (27)in a more compact form:

Sθδqj,k+1

+Srδqj−1,k +Dmodδqj,k +Srδqj+1,k = RHSn

j,k

+Sθδqj,k−1,

(29)

whereDmod = I/δt + Dr + Dθ.This equation gives rise to at least four different types of solution procedures:

(1) Classical explicit methods are very special cases in which the sub- and super-diagonal block matrices together withDr andDθ are neglected. The only ma-trix to be retained here is (1/δt)× (the identity matrix), i.e., the first term onthe LHS of Eq. (28). This yields the vector equation (see M5/Fig. 1):

[Iδt

]δqj,k = RHSnj,k. (30)

(2) Semi-explicit methods are obtained by preserving the diagonal entries,dj,k,

of the block diagonal matrixDmod (see M4/Fig. 1). This method has beenverified to be numerically stable even when large Courant-Friedrichs-Levy(CFL) numbers are used. In particular, this method is absolutely stable if theflow is viscous-dominated.

(3) Semi-implicit methods are recovered when neglecting the sub- and super-diagonal block matrices only, but retaining the block diagonal matrices (seeM3/Fig. 1). In this case the matrix equation reads:

Dmodδqj,k = RHSnj,k. (31)

We note that invertingDmod is a straightforward procedure, which can bemaintained analytically or numerically.

(4) A fully implicit solution procedure requires retainingall the block matrices onthe LHS of Eq. (28). This yields a global matrix that is highlysparse (M1/Fig.1). In this case, semi-direct methods such as the “Approximate FactorizationMethod” (AFM: Beam, Warming, 1978) and the “Line Gauss-Seidel Relax-ation Method” (-LGS: MacCormack, 1989) are considered to beefficientpreconditioners for solving the set of radiative MHD-equations within thecontext of defect-correction iteration method (see Hujeirat (2005) and the ref-erences therein). Furthermore, Krylov sub-iterative methods may prove to bemore efficient and robust than the above-mentioned semi-direct methods.

In the case that only stationary solutions are sought, convergence to steady state canbe accelerated by adopting the so-called “Residual Smoothing Method” (Hujeirat,

17

2005) This method is based on associating a time step size with the local CFL-number at each grid point. While this strategy is efficient at providing quasi-stationarysolutions within a reasonable number of iterations, it is incapable at providing phys-ically meaningful time scales for features that possess quasi-stationary behaviour.Here we suggest to use the obtained quasi-stationary solutions as initial configu-ration and re-start the calculations using a uniform and physically relevant timestep.

5 Numerical techniques

In this section we briefly describe several algorithmic steps for solving the continu-ity equation and the generalization procedure.

Fig. 3. The location of the variables using the staggered grid discretization.

(1) The continuity equation is discretized using the staggered grid strategy withinthe context of finite volume philosophy (Fig. 3).

1√−g

∂

∂r

(

√

−g/grr DU)∣

∣

∣

j,k

⇒[√−g/grr U

−→Dr ]

r j+1r j

[∫ √−gdr]

r j+1r j

∣

∣

∣

∣

∣

k

=[√−g/grr U

−→Dr ]r=r j+1 − [

√−g/grr U

−→Dr ]r=r j

[∫ √−gdr]

r j+1r j

∣

∣

∣

∣

∣

k

, (32)

1√−g

∂

∂θ

(

√

−g/gθθ)

DV∣

∣

∣

j,k

⇒[√−g/gθθV

−→Dθ]θk+1θk

[∫ √−g/gθθdθ]

θk+1θk

∣

∣

∣

∣

∣

∣

j

=[√−g/gθθV

−→Dθ]θk+1 − [

√−g/gθθV

−→Dθ]θk

[∫ √−gdθ]θk+1

θk

∣

∣

∣

∣

∣

∣

j

, (33)

18

where

−→Dr

j,k =

D j,k + f rj,k if U j,k < 0

D j−1,k + f rj−1,k if U j,k ≥ 0

;−→Dθ j,k =

D j,k + f θj,k if V j,k < 0

D j,k−1 + f θj,k−1 if V j,k ≥ 0(34)

The functionsf r and f θ are corrections for maintaining higher order spatialaccuracies.

(2) In order to obtain second order temporal accuracy, we write the continuityequation as follows:

δDδt+ ϑL1n+1(D) + (1− ϑ)L1n(D) = 0, (35)

whereϑ denotes the parameter of the stabilized Crank-Nicolson method forachieving second order temporal accuracy.L1h resembles the advection oper-ator at the new time level (n+1) and the old time level (n) andδD = Dn+1−Dn.

Taylor-expanding the variableDn+1 in time and considering first order termsonly, the continuity equation gets the following form:

δDδt+ ϑL1n+1(δD) + L1n(D) = 0, (36)

(3) Define the defectdj,k at every grid point:

dj,k = −δDδt− ϑL1n+1

high(D) − (1− ϑ)L1nhigh(D)| j,k, (37)

where the subscript ”high” means that the transport operators are evaluatedusing a spatially accurate advection scheme.

(4) Define at each grid point the following operator:

LD = −δDδt− ϑL1n+1

low (D) − (1− ϑ)L1nhigh(D)| j,k. (38)

Compute the following entries at each grid point:

Srj,k =

∂LD∂D j−1,k

, D j,k =∂LD∂D j,k

, Srj,k =

∂LD∂D j+1,k

,

Szj,k =

∂LD∂D j,k−1

, Szj,k =

∂LD∂D j,k+1

(39)

(5) In the one-dimensional case, the following matrix equation should be solvedat each grid point:

(

S j,k, D j,k, S j,k

)

δD j−1,k

δD j,k

δD j+1,k

=

d j−1,k

d j,k

d j+1,k

, where j=1, J, and k= const.(40)

19

For J number of grid points in the radial direction, this yields the tri-diagonalmatrix equation:

D1 S1

S2 D2 S2

. . .. . .. . .

SJ DJ

δD1

δD2

...

δDN

=

d1

d2

...

dN

for constant k (41)

Although this matrix equation is linear in D, it should be solved iteratively torecover the high spatial accuracy on the right hand side.

Similarly, if the continuity and the radial momentum equation are to besolved in one dimension as a coupled system, we may obtain thefollowingrelation at each grid point:

L11j−1 L12j−1 L11j L12j L11j+1 L12j+1

L21j−1 L22j−1 L21j L22j L21j+1 L22j+1

δD

δM

j−1

δD

δM

j

δD

δM

j+1

=

dD

dM

j−1

dD

dM

j

dD

dM

j+1

(42)

for j = 1, J and k= const.,

whereLmnl =∂Lm∂qn| j=l . Specifically, L1 is the density equation and L2 the

momentum equation.For J number of points this yields a tri-diagonal block matrix, in which eachblock has the dimension 2× 2.For a given set of equations in one dimension, we have just to replace theabove 2× 2 block matrix by a square block matrix whose dimensions areNeq× Neq, whereNeq is the number of unknown variables:

� �

� � �

. . .. . .. . .

� �

δq1

δq2

...

δqJ

=

d1

d2

...

dJ

. (43)

q here is a vector ofNeq entries.

20

The extension into two-dimensions gives rise to a matrix equation of thefollowing form:

A

δq1,1

...

δqJ,1

δq1,2

...

δqJ,K

=

d1,1

...

dJ,1

d1,2

...

dJ,K .

(44)

The matrixA has a similar structure as M1 in Fig. 1. This matrix equation issolved iteratively, using a non-direct inversion procedure.

6 Test calculations

The verification tests of the Newtonian version of the present algorithm have beenpresented in a series of papers (see Hujeirat (2005) and the references therein).Nevertheless the modifications made here are serious and deserve appropriate testcalculations to ensure bug-free runs as well as a consistentre-production of theresults in the Newtonian regime.In the following we briefly mention several of the test calculations performed:

• The shock tube problem - STPIn the case of low fluid-velocities, the modification made should enable capturingof shocks propagating at sub-relativistic speeds, irrespective of the accuraciesused. Therefore, we have applied the algorithm to the well-known Sod-problem(see Hujeirat (1995) and the references therein). Fig. 4 shows that the algorithmis indeed capable of re-producing Sod’s solution with high accuracy.

• The ultra-relativistic shock tube problemThe speed of the shocks in the Sod’s problem can be made arbitrary large, de-pending on the initial ratio of the pressure in the tube. While non-relativisticsolvers may produce propagating velocities that exceeds the speed of light, aconservative and accurate relativistic solver should produce velocities that canbe extremely close to but never exceed the speed of light.In Fig. 5 the one-dimensional profiles of the density, velocity, temperature andLorentz factorut are displayed. These profiles agree qualitatively with the ana-lytical solution of the relativistic STP provided by Marti and Muller (2003). In aforthcoming paper, we intend to quantitatively compare theprofiles for extremelylarge Lorentz factors.

Fig. 5 demonstrates the strong robustness of the algorithm and its capabil-

21

ity to capture the propagation of extreme ultra-relativistic shocks in which theLorentz factor is of order 1000. Such robustness is essential to enable modelingjetted Gamma-Ray bursts, where the Lorentz factors are in the excess of severalhundreds.

Fig. 4. The classical non-relativistic shock tube problem.The profiles of the density, ve-locity, temperature and pressure are displayed. The advection scheme used here is secondorder in time and third order in space. 200 uniformly distributed finite volume cells areused.

• Relativistic Bondi accretion onto Schwarzschild black holesThis problem is appropriate for testing the capability of the solver at treatingtransonic stationary accretion flows onto Schwarzschild black holes, assumingperfect spherical symmetry. This problem has been investigated by several au-

22

Fig. 5. The ultra-relativistic shock tube problem. The radial distributions of the density,radial velocity, temperature and the modified Lorentz factor U t are shown. The accuracy ofthe scheme and the number of points are identical to those in Fig. 4. This calculation showsthat shocks propagating with Lorentz factors of order 1000 can be safely treated with ouralgorithm.

thors (see Michel (1972), also see Hawley et al. (1984a,b) for a comprehensivedescription of the numerical treatment). In this problem, aconstant flux of anideal gas is said to be accreted by a non-rotating black hole.Depending on lo-cation of the outer boundary and on the temperature of the flow, the initiallysubsonic inflow should make a transition into the super-sonic regime at a spe-cific radius, which appears to be determined entirely by the constant of motion.On the other hand, the Lorentz factor of the flow as it crosses the inner boundaryshould approach the speed of light, depending on how close the inner radius is

23

to the event horizon. In Fig. 6 we display the radial distributions of the velocity,density, temperature, Lorentz factor and the Mach number, which clearly well-agree with the known analytical solutions. In obtaining these results we used apseudo time-stepping scheme to enhance convergence. The very last time stepsize in this calculations corresponds to Courant number 2000, approximately.

• Standing shocks around black holesThe purpose of this test is mainly to examine the capability of the algorithm at re-producing the formation of the two-dimensional curved standing shocks arounda Schwarzschild black hole that have been obtained using theNewtonian versionof the algorithm.

This problem is similar to the forward facing step in computational fluid dy-namics. Here a cold and dense disk has been placed in the innermost equatorialregion: [1≤ r ≤ 10] × [−0.3 ≤ θ ≤ 0.3] (see Figures 7 and 8). Vanishing in-and out-flow conditions have been imposed at the boundaries of the cold disk.The gas surrounding the disk is taken to be inviscid, thin, hot and non-rotating.The cold disk here serves as a two-dimensional barrier that disturbs the gas fromotherwise a spherically symmetric freely falling flow onto aSchwarzschild blackhole and, instead, it forces the inflow to form a curved shock which eclipses thecold disk.

In solving the HD-equations, an advection scheme of third order spatial accu-racy and of first order accuracy in time has been used.

Hence the scheme is taken to be highly diffusive in time in order to damp os-cillations and to accelerate convergence into steady-state. The domain of calcu-lation is sub-divided into 200 strongly-stretched finite volume cells in the radialand 60 in the horizontal direction. In Fig. 8 the 1D radial andhorizontal pro-files, the 2D configuration of the density, temperature and the velocity field areshown. Indeed, the algorithm shows that it is numerically stable and capable ofcapturing steady-state shocks with complicated shock structures even for largeCFL-numbers.

7 Summary

In this paper we have extended the previous Newtonian implicit algorithm to enablesolving the hydrodynamical equations in general relativity. The 3D axi-symmetrichydrodynamical equations have been presented in the background of a Kerr metricof a black hole using the Boyer-Lindquist coordinates. The equations have beenformulated in conservative form and subsequently solved numerically, using thefinite volume formulation. The new extension can be well accommodated withinthe hierarchical solution scenario, in which the degree of implicitness can be madedynamical, depending on the hydrodynamical problem in hand. In particular, formodeling strongly time-dependent astrophysical flows, such as moving shocks, thepre-conditioners used are tri-diagonal matrices that are solved successively. Al-

24

though the computational costs per time step may be one orderof magnitude largerthan their explicit counterparts, this can be compensated through a reduction of theoverall number of time steps required to recover a physically reliable time scale.

On the other hand, the efficiency and robustness of the HSS are superior, if thesolutions sought are stationary or quasi-stationary, irrespective of whether the flowis dissipative or not.

Finally, a unification scheme for various numerical methodshas been presented. Inparticular, the HSS algorithm enables the construction of alarge variety of solvers,in which the degree of implicitness may range from purely explicit up to stronglyimplicit, depending on the physical properties of the underlying flow problem.Thus, the HSS is actually a unified algorithm for treating weakly compressible, in-compressible, time-dependent, time-independent, radiative, magnetized non-dissi-pative or strongly dissipative flows. As a consequence, using the HSS algorithm,we are able to save a large number of working hours which otherwise would go indesigning different solvers for different physical problems.In a subsequent paper, we intend to discuss and describe the inclusion of the mag-netohydrodynamical equations in general relativity into the present solver.

Acknowledgment A.H. thanks Prof. J. Dusek for reading the manuscript and forthe hospitality during his visit to the Institut de mechanique des fluides et dessolides, CNRS and the Louis Pasteur University in Strasbourg. This work hasbeen partially supported by the Klaus-Tschira Stiftung under the project number00.099.2006.

References

Aloy, M.-A., Ibanez, J.M., Mart, J.M., Muller, E., 1999, ApJS, 122, 151 (GENE-SIS)

Anninos, P., Fragile, P. C., 2003, ApJ. Suppl. Ser., 144, Iss. 2, 243-257Anninos, P., Fragile, P. C., Salmonson, J.D., 2005, ApJ, 635, 723Baiotti, L., Hawke, I., Montero, P.J., Rezzolla, L., 2003, MSAIS, 1, 210Banyuls, F., Font, J. A.,Ibanez, J. M., Martı, J. M., Miralles, J. A., 1997, ApJ, 476,

221Beam, R.M., Warming, R.F., 1978, AIAA, 16, 393Bodenheimer, P., Tohline, J. E., Black, D. C., 1978, BAAS, 10, 655Bohmer, K., Stetter H. J., 1984, “Defect Correction Methods: Theory and Applica-

tions”, Springer–Verlag, Wien-New YorkClarke, D.A., 1996, APJ, 457, 291Del Zanna, L., Bucciantini, N., 2002, A&A, 390, 1177-1186Del Zanna, L., Zanotti, O., et al., 2007, A&A, 473, 11De Villiers, J.-P., Hawley, J.F., 2003, ApJ, 589, 458Enander, R., 1997, SIAM J. Sc. Comp., 18, 5, 1243

25

Eulderink, F., Mellema, G., 1995, Astron. Astophys. Suppl.Ser., 110, 587Falle, S.A.E.G., 2003, astro-ph/0308396Fletcher, C.A.J., 1988, ”Computational Techniques for Fluid Dynamics”, Vol. I, II,

Springer-Verlag, LondonFont, J.A., Miller, M., Suen, W., Tobias, M., 2000, Phys. Rev. D, 61, 044011Font, J.A., 2003, LRR, 6, 4Fryxell, B. et al., 2000, ApJS, 131, 273-334 (FLASH)Gammie, C. F., McKinney, J. C., Toth, G., 2003, ApJ, 589, 444-457 (HARM)Gardiner, T.A., Stone, J.M., 2006, ASPC, 359, 143Hackbusch, W., 1994, “Iterative Solution of Large Sparse Systems of Equations”,

Springer–Verlag, New York-Berlin-HeidelbergHirsch, C., 1988, ”Numerical Computation of Internal and External Flows”, Vol. I,

II, John Wiley & Sons, New YorkHujeirat, A., 1995, A&A,295, 268Hujeirat, A., Rannacher, R., 2001, New Ast. Reviews, 45, 425Hujeirat, A., 2005, CoPhC, 168, 1Hawley, J. F., Smarr, L. L., Wilson, J. R., 1984a, ApJ, 277, 296-311Hawley, J. F., Smarr, L. L., Wilson, J. R., 1984b, ApJS, 55, 211-246Koide, S., Shibata K., Kudoh, T., 1999, ApJ, 522, 727Komissarov, S.S., 2004, MNRAS, 350, 1431Liebendorfer, M., Rosswog, S., Thielemann, F.-K., 2002, ApJS, 141, 229LMacCormack, R., 1989, Computers and Fluids, 17, 135Marti, J.M., Muller, E., 2003, LRR, 6, 7Michel, F.C., 1972, Ap&SS, 15, 163Mignone, A., Bodo, G., 2003, NewAR, 47, 581Mignone, A., Bodo, G., Massaglia, S., Matsakos, T., Tesileanu, O., Zanni, C., Fer-

rari, A., 2007, ApJS, 170, 228-242 (PLUTO)Meliani, Z., Keppens, R., et al., 2007, MNRAS, 376, 1189Mizuno, Y., Nishikawa, J.-I., et al., 2006, astro-ph/0609004 (RAISHIN)Nagel, W.E., Jager, W., Resch, M. (Eds.), 2006, ”High Performance Computing in

Science and Engineering ’06”, Springer, Berlin, Heidelberg, New York, 2006O’Shea, B.W., Bryan, G., Bordner, J., Norman, M. L., Abel, T., Harkness, R., Krit-

suk, A., 2004, astro-ph/0403044 (ENZO)Richtmyer, R.D., Morton, K.W., 1967, ”Difference methods for initial value prob-

lems”, 2nd ed., Interscience, New YorkShibata, M., Sekiguchi, Y., 2005, PhRvD, 72, 4, 044014Spindeldreher, S., PhD thesis, 2002, Ruperto-Carola University of Heidelberg, Ger-

manyStetter, H.J., 1978, Numer. Math., 29, 425–443Stone, J.M., Norman, M.L., 1992, ApJS, 80, 753Swanson, R.C., Turkel, E., 1997, NASA,TP-3631, 81Swesty, F.D., 1995, ApJ, 445, 811Tassoul, J.-L., “Theory of rotating stars”, PUP, PricetonToth, G., Keppens, R., Botchev M.A., 1998, A&A, 332, 1159 (VAC)Wang, P., Abel, T., Zhang, W., 2007, astro-ph/0703742 (ENZO)

26

Wilson, J. R., 1972, ApJ, 173,431Wuchterl, G., 1990, A&A, 238, 83Zhang, W., MacFadyen, A. I., 2006, ApJS, 164, 255-279 (RAM)Ziegler, U., 1998, Comp. Phys. Comm., 109, 111 (NIRVANA)

27

Fig. 6. The Bondi accretion problem onto a Schwarzschild black hole. The profiles of theradial velocity, the relativistic density, temperature, the Lorentz factorU t and the radialMach number. In the lower-right panel profiles of the Courantnumber (solid line) and thecorresponding residual (dashed line) versus the iterationnumber are displayed. Althoughthe problem is spherically symmetric, the calculations have been carried out using 200grid points in the radial and 30 in the horizontal-direction. The accuracy of the advectionscheme is set to be first order in time and third order in space.This test enables us toexamine the capability of the algorithm at capturing steadysolutions that are essentiallyone-dimensional using a 2D numerical scheme. We have verified that the 30-profiles in theradial direction obtained at differentθ are identical to machine- accuracy.

28

Fig. 7. The 2D distribution of the temperature (in units of 109K) of a freely falling non-rela-tivistic gas onto a Schwarzschild black hole surrounded by astatic cold disk (top panel). Inthis figure, color gradients run as follows: red color corresponds to large temperature-val-ues, green to intermediate and blue to low values. The distribution in the second and thirdpanels have been obtained using the general relativistic version of the algorithm. Here theinflowing matter across the outer boundary has the temperatures 109K (middle) and 1010K(bottom).

29

Fig. 8. Distribution of the grid points used in the calculations. A strong non-uniform dis-tribution of the grid points has been constructed to enable accurate capturing of standingshocks surrounding the cold disk. The tensor-product mesh consists of 275 finite volumecells in the radial and 130 in the horizontal direction, respectively. In the lower panel theprofiles of the density, temperature, radial and vertical velocities along the equator andhorizontal along the constant radiusr = 2.5 are shown.

30